modular arithmetic reasoning

Solving multi-modular integer constraint systems—comprising polynomial equalities and inequalities under distinct moduli—is notoriously difficult in cryptographic protocol verification; existing SMT solvers fail to exploit their inherent algebraic structure. Method: This paper introduces the first resolution-based decision procedure tailored for multi-modular reasoning. Contributions/Results: (1) Constraints are partitioned by modulus, and novel algebraic lifting/reduction mechanisms enable information sharing across modular subsystems; (2) Weighted Gröbner basis theory is integrated into the SMT framework for precise multi-modular algebraic reasoning—the first such incorporation; (3) A modular, embeddable solving pipeline is constructed. Evaluated on Montgomery multiplication and zero-knowledge proof implementation verification, our method substantially outperforms state-of-the-art SMT solvers: solution success rate improves by 42%, and average verification time decreases by a factor of 5.8.

This work addresses the satisfiability problem for Constrained Horn Clauses (CHCs) over the theory of fixed-size bit-vectors (𝒯_B), a key challenge in bit-precise program verification. The paper introduces Mosaic, a novel framework that enables modular cooperative reasoning between bit-vector and integer arithmetic theories for the first time. By decomposing CHC problems into theory-specific fragments and facilitating cross-theory translation and information exchange, Mosaic overcomes the scalability limitations of existing CHC solvers in bit-level verification tasks. Implemented on top of Z3 and Spacer, Mosaic demonstrates substantial performance improvements over native Spacer on benchmarks involving bit-vector operations, thereby validating its effectiveness and superiority.

Verifying large-scale arithmetic circuits for wide-word operations often incurs prohibitive computational costs due to reliance on arbitrary-precision integer arithmetic, which scales poorly with word length. This work proposes a hybrid algebraic verification approach based on polynomial reasoning that integrates both linear and nonlinear rewriting strategies. Crucially, it introduces— for the first time—a parallel multimodal homomorphic image technique that performs algebraic reasoning simultaneously over multiple prime moduli, thereby entirely eliminating the need for large-integer computations. Implemented in the TalisMan2.0 tool, the method demonstrates significant performance advantages over existing verification schemes on multiplier benchmarks, offering both high efficiency and strong scalability.

Formal verification of compilers incurs high maintenance costs, especially when modifications necessitate extensive re-verification. Method: This paper introduces the first trusted rewriting engine framework for Coq, modeling compilers as collections of algebraic rewrite rules—each independently verifiable. It employs theorem-driven modeling, metaprogramming-based automated synthesis, and proof reuse to enable rule-level formal verification and automatic composition. Contribution/Results: The framework decouples rule verification from compiler construction, significantly reducing verification and maintenance overhead. Evaluated in the Fiat Cryptography toolchain, the generated command-line compiler achieves approximately 1000× speedup over prior verified counterparts. Moreover, its proofs are more concise and exhibit substantially higher reusability across compiler transformations.

Satisfiability checking for non-polynomial integer arithmetic—particularly involving exponential functions—remains a major challenge in program verification; existing SMT solvers support only polynomial constraints and lack systematic handling of exponential integer arithmetic. Method: This paper presents the first sound, complete, and practical SMT solver for exponential integer arithmetic. It introduces an incremental linearization framework tailored to exponential semantics, constructs exact cutting planes leveraging both convexity and discreteness, and designs a semantic-lemma-driven spurious-counterexample elimination mechanism. The solver integrates conflict-driven clause learning (CDCL), integer programming, and interval propagation for efficient reasoning. Results: Experimental evaluation on diverse program verification benchmarks shows that our approach achieves over a 3× improvement in solving success rate and reduces average solving time by 62%, significantly outperforming state-of-the-art tools.

This work addresses the limitations of MCSat in solving complex SMT problems involving nonlinear integer and real arithmetic by reformulating it as a theory-agnostic proof system. By formally capturing key implementation mechanisms from the Yices2 solver, the authors derive a unified and general framework of MCSat inference rules, which they instantiate across multiple theories—including propositional logic, nonlinear real arithmetic, and uninterpreted functions. This approach not only integrates core design choices of modern SMT solvers but also establishes the first unified MCSat calculus supporting multiple theories, substantially enhancing its expressiveness and applicability. The effectiveness of the proposed framework is demonstrated through representative examples.

Existing formal verification methods struggle to effectively validate correctness properties of programs characterized by symmetry. This paper introduces the first Hoare logic framework specifically designed for symmetry verification: it replaces conventional pre- and postconditions with group actions, defines a formal syntax for group-action specifications, and establishes a natural entailment relation—enabling rigorous, compositional reasoning about symmetry properties of imperative programs. The approach integrates group action theory, Hoare logic, and static analysis, and is implemented in the prototype tool SymVerif. Evaluation on multiple hand-crafted benchmarks confirms the framework’s effectiveness; notably, it uncovered a logical inconsistency in the symmetry formulation of an existing dynamical systems model. The core contribution is the first sound and complete Hoare logic for symmetry based on group actions—establishing a provably correct, implementable paradigm for symmetry-driven program verification.

This work addresses the high degree of manual effort and tediousness inherent in existing automated reasoning algorithms—such as those for hyper-exponential quantifier elimination—for complexity analysis. The paper proposes a higher-order abstract interpretation framework grounded in operator semantics, which automatically abstracts symbolic programs into numerical recurrence relations. By integrating termination analysis, fixed-point theory, and SMT solving techniques, the method enables fully automated derivation and verification of asymptotic upper bounds on computational complexity. This approach substantially reduces human intervention while significantly enhancing the automation, efficiency, and scalability of complexity analysis for intricate algorithms.

This study addresses the problem of systematically constructing logical systems corresponding to program abstractions to support formal reasoning. By associating logical systems with finite abstract domains, the work proposes a general method: for a given abstract lattice, it constructs a logic whose Lindenbaum–Tarski algebra is isomorphic to the abstraction, and derives corresponding axioms and inference rules. This approach establishes, for the first time, a systematic connection between abstract interpretation and proof theory as well as algebraic logic, enabling logical modeling of non-Cartesian abstractions such as octagons. The resulting logical connectives and inference systems preserve the concretization map and, under suitable conditions, satisfy soundness and completeness. The framework naturally extends to Cartesian products, multi-variable settings, and non-Cartesian abstract domains.